Yes, you can change the price of your app whenever you want for whatever you want. However, the price change will affect everyone who 'buys' your app.
There are hundreds, maybe thousands, of apps that schedule price changes on a regular basis. I know 1 app that does a free-to-.99 price change every 2 weeks. Apple doesn't care but I'm pretty sure the app customers do care.
EDIT: As asked in the comments, here's a clarification. I'm referring to the price people pay for the app in the App Store. For example, you make the app free for everyone in the App Store for 2 months. You then decide to change the app price to $1.99. That means everyone who downloads the app now has to pay $1.99. If you got the app for free then you get all bug-fix updates to that app for free as long as the developer doesn't create a new SKU for the updates.
We've all seen this happen recently when Tweetbot updated their app to a new version and created a new SKU for it. To Apple, and your device, this is a different app. If you paid for the first version you need to pay again for the update because the dev and Apple consider it a different app.
If you want to make the app free and then have an in-app purchase to upgrade and get additional features, that's acceptable. In fact, that's the preferred way to do it and many apps to that. I hope that